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## G = C3×C32⋊D8order 432 = 24·33

### Direct product of C3 and C32⋊D8

Aliases: C3×C32⋊D8, C331D8, C32⋊(C3×D8), C6.21S3≀C2, D6⋊S31C6, C322C81C6, (C32×C6).3D4, C2.3(C3×S3≀C2), (C3×C6).3(C3×D4), (C3×D6⋊S3)⋊8C2, (C3×C322C8)⋊3C2, C3⋊Dic3.5(C2×C6), (C3×C3⋊Dic3).31C22, SmallGroup(432,576)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C3×C32⋊D8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C3×C3⋊Dic3 — C3×D6⋊S3 — C3×C32⋊D8
 Lower central C32 — C3×C6 — C3⋊Dic3 — C3×C32⋊D8
 Upper central C1 — C6

Generators and relations for C3×C32⋊D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 492 in 96 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4, C22 [×2], S3 [×2], C6, C6 [×12], C8, D4 [×2], C32, C32 [×4], Dic3 [×2], C12, D6 [×2], C2×C6 [×6], D8, C3×S3 [×8], C3×C6, C3×C6 [×6], C24, C3⋊D4 [×2], C3×D4 [×2], C33, C3×Dic3 [×2], C3⋊Dic3, S3×C6 [×6], C62 [×2], C3×D8, S3×C32 [×2], C32×C6, C322C8, D6⋊S3 [×2], C3×C3⋊D4 [×2], C3×C3⋊Dic3, S3×C3×C6 [×2], C32⋊D8, C3×C322C8, C3×D6⋊S3 [×2], C3×C32⋊D8
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, C2×C6, D8, C3×D4, C3×D8, S3≀C2, C32⋊D8, C3×S3≀C2, C3×C32⋊D8

Permutation representations of C3×C32⋊D8
On 24 points - transitive group 24T1318
Generators in S24
(1 12 21)(2 13 22)(3 14 23)(4 15 24)(5 16 17)(6 9 18)(7 10 19)(8 11 20)
(1 12 21)(2 13 22)(3 23 14)(4 24 15)(5 16 17)(6 9 18)(7 19 10)(8 20 11)
(1 12 21)(2 22 13)(3 23 14)(4 15 24)(5 16 17)(6 18 9)(7 19 10)(8 11 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 24)(18 23)(19 22)(20 21)

G:=sub<Sym(24)| (1,12,21)(2,13,22)(3,14,23)(4,15,24)(5,16,17)(6,9,18)(7,10,19)(8,11,20), (1,12,21)(2,13,22)(3,23,14)(4,24,15)(5,16,17)(6,9,18)(7,19,10)(8,20,11), (1,12,21)(2,22,13)(3,23,14)(4,15,24)(5,16,17)(6,18,9)(7,19,10)(8,11,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,24)(18,23)(19,22)(20,21)>;

G:=Group( (1,12,21)(2,13,22)(3,14,23)(4,15,24)(5,16,17)(6,9,18)(7,10,19)(8,11,20), (1,12,21)(2,13,22)(3,23,14)(4,24,15)(5,16,17)(6,9,18)(7,19,10)(8,20,11), (1,12,21)(2,22,13)(3,23,14)(4,15,24)(5,16,17)(6,18,9)(7,19,10)(8,11,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,24)(18,23)(19,22)(20,21) );

G=PermutationGroup([(1,12,21),(2,13,22),(3,14,23),(4,15,24),(5,16,17),(6,9,18),(7,10,19),(8,11,20)], [(1,12,21),(2,13,22),(3,23,14),(4,24,15),(5,16,17),(6,9,18),(7,19,10),(8,20,11)], [(1,12,21),(2,22,13),(3,23,14),(4,15,24),(5,16,17),(6,18,9),(7,19,10),(8,11,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,16),(17,24),(18,23),(19,22),(20,21)])

G:=TransitiveGroup(24,1318);

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4 6A 6B 6C ··· 6H 6I ··· 6X 8A 8B 12A 12B 24A 24B 24C 24D order 1 2 2 2 3 3 3 ··· 3 4 6 6 6 ··· 6 6 ··· 6 8 8 12 12 24 24 24 24 size 1 1 12 12 1 1 4 ··· 4 18 1 1 4 ··· 4 12 ··· 12 18 18 18 18 18 18 18 18

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 C3×D8 S3≀C2 C32⋊D8 C3×S3≀C2 C3×C32⋊D8 kernel C3×C32⋊D8 C3×C32⋊2C8 C3×D6⋊S3 C32⋊D8 C32⋊2C8 D6⋊S3 C32×C6 C33 C3×C6 C32 C6 C3 C2 C1 # reps 1 1 2 2 2 4 1 2 2 4 4 4 8 8

Matrix representation of C3×C32⋊D8 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 5 3 5 3 3 5 2 3 0 0 1 0 0 0 0 4
,
 3 2 4 3 4 5 5 6 3 3 6 1 0 0 0 1
,
 4 2 1 4 6 6 0 3 2 5 6 4 1 1 6 5
,
 0 2 4 1 2 4 6 5 3 3 3 1 6 1 4 0
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[5,3,0,0,3,5,0,0,5,2,1,0,3,3,0,4],[3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[4,6,2,1,2,6,5,1,1,0,6,6,4,3,4,5],[0,2,3,6,2,4,3,1,4,6,3,4,1,5,1,0] >;

C3×C32⋊D8 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes D_8
% in TeX

G:=Group("C3xC3^2:D8");
// GroupNames label

G:=SmallGroup(432,576);
// by ID

G=gap.SmallGroup(432,576);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,197,1011,514,80,4037,3036,362,1189,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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